A225957 - OEIS

%I #6 May 21 2013 23:58:01

%S 1,2,-6,12,38,-108,148,168,-922,2294,-2656,-1732,17908,-44516,60896,

%T -6936,-206474,650848,-1181394,1146324,865832,-6609592,16632596,

%U -26643544,22498916,23275482,-144152248,349896736,-563311472,532552508,233516176,-2378435472,6264582710

%N O.g.f.: exp( Sum_{n>=1} -(sigma(2*n^3) - sigma(n^3)) * (-x)^n/n ).

%C Compare to the Jacobi theta_3 function:

%C 1 + 2*Sum_{n>=1} x^(n^2)  =  exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*(-x)^n/n ).

%C Here sigma(n) = A000203(n), the sum of the divisors of n.

%H Paul D. Hanna, <a href="/A225957/b225957.txt">Table of n, a(n) for n = 0..1000</a>

%F O.g.f.: exp( Sum_{n>=1} -A054785(n^3)*(-x)^n/n ).

%e O.g.f.: A(x) = 1 + 2*x - 6*x^2 + 12*x^3 + 38*x^4 - 108*x^5 + 148*x^6 + 168*x^7 +...

%e where

%e log(A(x)) = 2*x - 8*x^2/2 + 26*x^3/3 - 32*x^4/4 + 62*x^5/5 - 104*x^6/6 + 114*x^7/7 - 128*x^8/8 + 242*x^9/9 - 248*x^10/10 + 266*x^11/11 - 416*x^12/12 +...+ -(-1)^n*A054785(n^3)*x^n/n +...

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, -(sigma(2*m^3)-sigma(m^3))*(-x)^m/m)+x^2*O(x^n)), n)}

%o for(n=0, 50, print1(a(n), ", "))

%Y Cf. A225958, A225959, A054785, A000203; variants: A215603, A225925, A195584.

%K sign

%O 0,2

%A _Paul D. Hanna_, May 21 2013